The first rule of doing physical calculations
on paper is that the numbers being manipulated don't stand alone:
most quantities have attached dimensions, to be ignored at our
peril. As computations become more complex, keeping track of
dimensions is what keeps us from inadvertently assigning a mass to
what should be a length or adding acceleration to velocity — it
establishes a type system for numbers.

Manual checking of types is tedious, and as a result, it's also
error-prone. When human beings become bored, their attention
wanders and they tend to make mistakes. Doesn't type checking seem
like the sort of job a computer might be good at, though? If we
could establish a framework of C++ types for dimensions and
quantities, we might be able to catch errors in formulae before
they cause serious problems in the real world.

Preventing quantities with different dimensions from interoperating
isn't hard; we could simply represent dimensions as classes that
only work with dimensions of the same type. What makes this
problem interesting is that different dimensions can be combined,
via multiplication or division, to produce arbitrarily complex new
dimensions. For example, take Newton's law, which relates force to
mass and acceleration:

F = ma

Since mass and acceleration have different dimensions, the
dimensions of force must somehow capture their combination. In
fact, the dimensions of acceleration are already just such a
composite, a change in velocity over time:

dv/dt

Since velocity is just change in distance (l) over time (t),
the fundamental dimensions of acceleration are:

(l/t)/t = l/t2

And indeed, acceleration is commonly measured in "meters per second
squared." It follows that the dimensions of force must be:

ml/t2

and force is commonly measured in kg(m/s2), or
"kilogram-meters per second squared." When multiplying quantities
of mass and acceleration, we multiply their dimensions as well and
carry the result along, which helps us to ensure that the result is
meaningful. The formal name for this bookkeeping is dimensional
analysis, and our next task will be to implement its rules in the C++
type system. John Barton and Lee Nackman were the first to show
how to do this in their seminal book, Scientific and Engineering
C++[BN94]. We will recast their approach here in
metaprogramming terms.